12. Write the formula for the nth term of the geometric sequence: 3, 12, 48,

The given geometric sequence is: 3, 12, 48, ...

To find the formula for the nth term of a geometric sequence, you use the formula:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Where:
- \( a_n \) is the nth term,
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number.

### Step 1: Identify the first term (\(a_1\))
The first term \(a_1 = 3\).

### Step 2: Find the common ratio (\(r\))
The common ratio \(r\) can be found by dividing the second term by the first term:
\[
r = \frac{12}{3} = 4
\]

### Step 3: Write the nth term formula
Substitute the values of \(a_1\) and \(r\) into the formula:
\[
a_n = 3 \cdot 4^{(n-1)}
\]

So, the formula for the nth term of the geometric sequence is:
\[
a_n = 3 \cdot 4^{(n-1)}
\]

Would you like more details on any part of the process?

### Relative questions:
1. What is the 5th term of the sequence using the nth term formula?
2. How do you derive the sum of the first n terms in a geometric sequence?
3. What happens to the sequence if the common ratio changes?
4. How can you determine the common ratio if you only know the first and third terms of the sequence?
5. What is the difference between an arithmetic and a geometric sequence?

### Tip:
Always check the common ratio by dividing consecutive terms to ensure the sequence is indeed geometric before applying the nth term formula.

12. Write the formula for the nth term of the geometric sequence: 3, 12, 48, ...

Discover how to find the formula for the nth term of the geometric sequence: 3, 12, 48. This solution explains the steps to calculate the common ratio and derive the formula using the geometric sequence formula. Perfect for students in Grades 9-10.

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